Characterizing non-totally geodesic spheres in a unit sphere

A concircular vector field $\mathbf{u}$ on the unit sphere $\mathbf{S}^{n+1}$ induces a vector field $\mathbf{w}$ on an orientable hypersurface $M$ of the unit sphere $\mathbf{S}^{n+1}$, simply called the induced vector field on the hypersurface $M$. Moreover, there are two smooth functions, $f$ and $\sigma$, defined on the hypersurface $M$, where $f$ is the restriction of the potential function $\overline{f}$ of the concircural vector field $\mathbf{u}$ on the unit sphere $\mathbf{S}^{n+1}$ to $M$ and $\sigma$ is defined as $g\left(\mathbf{u}, N\right)$, where $N$ is the unit normal to the hypersurface. In this paper, we show that if function $f$ on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field $\mathbf{w}$ has a certain lower bound, then a characterization of a small sphere in the unit sphere $\mathbf{S}^{n+1}$ is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface $M$ in the direction of the vector field $\mathbf{w}$ with a non-zero function $\sigma$